They had this game for years at our local amusement park along with Punch-Out. Of course, if you played this game with a joystick, like Punch-Out! it would be ridiculous. What you can't see in the video is that the controller for this game was an actual arm you wrestled with, with force feedback. The video and audio was just there to make trying to beat the arm interesting.

Hate to break it to you, but I think that you're thinking of Arm Champs, which I have definitely seen around a fair bit. Arm Wrestling uses a joystick that moves only horizontally as well as a single button.

From the video, it appears to work on the same basic principles as Punch-Out -- watch for "tells" so that you can time your strikes to stun your opponent -- so I can imagine it being just as much fun to play. And I definitely appreciate the fact that someone at Nintendo said, "Hey, how can we make Punch-Out even more ludicrous than it already is? Maybe we could have a girl that shoots bullets out of her fingers and have you arm-wrestle a robot monkey who uses a magnet to pull on his own hand!" I can totally get behind that kind of thinking.